Inverse of a matrix

For a square matrix A, if

AB = BA = I

Then, B is the inverse of A

i.e. B = A ⁻¹

We will find inverse of a matrix by

• Elementary transformation
• Using adjoint 

Note: Since AB = BA = I

We can say B is the inverse of A.

i.e. B = A ⁻¹

 

We can also say,

A is the inverse of B

i.e. A = B ⁻¹

Thus, for inverse

We can write

  AA ⁻¹ = A ⁻¹ A = I

Where I is identity matrix of the same order as A

Let’s look at same properties of Inverse.

Properties of Inverse

1. For  a matrix A,
   A ⁻¹ is unique, i.e., there is only one inverse of a matrix
2. (A ⁻¹ ) ⁻¹   = A
3. (kA) ^-1 = 1/k A ^-1  

   Note: This is different from

   (kA) ^T = k A ^T

4. (A ^-1 ) ^T = (A ^T ) ^-1

5. (A + B) ^-1 = A ^-1 + B ^-1

6.  (AB) ^-1 = B ^-1 A ^-1